The so-called Lyashko–Looijenga covering (see [Looijenga1974]; [Vassiliev2002a]) is
a strong tool for constructing (or proving the existence of) the perturbations
of simple singularities with prescribed topological properties, such as
singularity types of different critical points, or intersection matrices of
vanishing cycles: see, e.g., [Lyashko1976]. The real version of this tool allows one
to construct and enumerate all topologically different Morsifications of real
simple singularities: see [Looijenga1978], [Chislenko1988], [Vassiliev2002b].

Many of these options are preserved for non-simple singularities: see
[Vassiliev2002a]; [Vassiliev2002b]. In particular, this method was used to predict the
existence of many Morsifications with prescribed properties and to describe their
expected topological characteristics, making it easy to give an explicit construction of these Morsifications. However, in this case this method is rather
experimental or heuristic, without clear guaranties that all perturbations
found by it actually exist. Therefore, it is important to track down the
restrictions of this method. Here are several explicit problems.

Let $f:({\mathbb{C}}^{n},0)\to({\mathbb{C}}^{1},0)$ be an isolated holomorphic function singularity,
$\mu$ be its Milnor number, $F(x,\lambda):({\mathbb{C}}^{n}\times{\mathbb{C}}^{\mu},0)\to({\mathbb{C}}^{1},0)$ be the miniversal deformation of $f$, and $\Sigma\subset{\mathbb{C}}^{\mu}$ be the complete
bifurcation set of functions of this deformation, i.e. the set of values of the
parameter $\lambda\in{\mathbb{C}}^{\mu}$ such that the corresponding function $f_{\lambda}=F(\cdot,\lambda)$ has less than $\mu$ different critical values at
critical points close to the origin. The Lyashko–Looijenga map sends any
point $\lambda$ from a small neighborhood $B_{\varepsilon}$ of the origin in
${\mathbb{C}}^{\mu}$ to the unordered collection of critical values of the function
$f_{\lambda}$ at points close to $0\in{\mathbb{C}}^{n}$ (or, which is equivalent but
sometimes more convenient, to the set of values of basic symmetric polynomials
of these critical values). If the singularity of $f$ is simple, then the
restriction of this map to $B_{\varepsilon}{\setminus}\Sigma$ is a local
covering over the configuration space $B(D,\mu)$ of all subsets of cardinality
$\mu$ in a very small (even with respect to $\varepsilon$) neighborhood $D$ of
the origin in ${\mathbb{C}}$: see [Looijenga1974]. In particular, any element $\alpha\in\pi_{1}(B(D,\mu))$ can be realized by a loop which can be lifted to a path in
$B_{\varepsilon}{\setminus}\Sigma$, covering this loop.

For non-simple singularities this is no longer the case. As previously, the
Lyashko–Looijenga map is submersive (and hence locally bijective) everywhere
in $B_{\varepsilon}{\setminus}\Sigma$ (this follows from the very notion of
miniversality). However, a sufficiently complicated path in $B(D,\mu)$,
lifted into ${\mathbb{C}}^{\mu}{\setminus}\Sigma$ in accordance with this local
bijectivity, can run out from the neighborhood of the origin in ${\mathbb{C}}^{\mu}$. This
is related to the fact that the Lyashko–Looijenga map for non-simple singularities is not proper: the preimage of the collection $(0,\dots,0)$ is the entire (positive-dimensional) $\mu=const$ stratum.11A
weaker substitute for the Lyashko–Looijenga covering theorem holds in the case
of parabolic singularities, if one writes the versal deformation in the
canonical monomial form and allows large travellings in the space ${\mathbb{C}}^{\mu}$: see
[Jaworski1986]. For more complicated singularities the situation is even worse.

Problem 1A.

Present explicit obstructions to
the Lyashko–Looijenga covering in terms of braid groups. Which braids
cannot be lifted to the space ${\mathbb{C}}^{\mu}{\setminus}\Sigma$?

Given a configuration of $\mu$ different points $z_{1},\dots,z_{\mu}$ in $D{\setminus}0$ and a system of non-intersecting paths connecting them with 0, any
perturbation $f_{\lambda}$ of $f$, which has these critical values, defines a Dynkin diagram: see [Arnold et al.1985], vol. 2. Any braid $l\in\pi_{1}(B(D,\mu))$
transforms this Dynkin diagram into another one in accordance with the
Picard–Lefschetz formulas: see [Arnold et al.1985] or [Vassiliev2002a]. If our braid $l$
can be lifted to a curve in $B_{\varepsilon}{\setminus}\Sigma$ starting at the
point $\lambda$ and covering this braid via the Lyashko–Looijenga map, then
the resulting Dynkin diagram is nothing but the Dynkin diagram of the
function $f_{\lambda^{\prime}}$ corresponding to the endpoint of this lifted curve and
defined by the same system of paths connecting the critical values with 0.

However, for complicated singularities, the number of Dynkin graphs, which can be obtained by the formal Picard-Lefschetz moves, is infinite, while the number of preimages of any non-discriminant configuration under the Lyashko–Looijenga map is bounded.

Problem 1B.

Let a non-simple singularity $f$ be given and the
Dynkin diagram of it be defined by an easily distinguished system of paths
connecting 0 with critical points of $f_{\lambda}$. Which Dynkin graphs can be
obtained from it by a sequence of formal Picard–Lefschetz moves defined by a
braid, but cannot appear as Dynkin diagrams of Morsifications $f_{\lambda^{\prime}}$
with the same critical values defined by the same system of paths?

Furthermore, for simple singularities, all partial collisions of $\mu$ critical
values can be realized, because the Lyashko–Looijenga map is proper. This
reduces the problem of enumeration of possible decompositions of the
initial critical point to a problem formulated in terms of Dynkin diagrams
and Picard–Lefschetz operators only: see [Lyashko1976]. Again, for non-simple
singularities, it is not the case. For example, any non-simple singularity
admits a system of paths connecting 0 with critical values, such that there are two vanishing cycles with the
intersection number equal $\pm 2$. Then we
surely cannot lift to $B_{\varepsilon}$ the collision of these two critical
values along these paths while keeping the remaining critical values unmoved.
Namely, the attempt to move these critical values towards one another by means
of the Lyashko–Looijenga submersion will throw the parameter $\lambda$ out of
any neighborhood of the origin in ${\mathbb{C}}^{\mu}$.

Problem 1C.

Are there more refined restrictions
to the collision of critical values? Is it true that for any two vanishing cycles, whose intersection
number is equal to $\pm 1$ or 0, we can lift
the collision of the corresponding critical values to $B_{\varepsilon}$ via the
Lyashko–Looijenga submersion?

In the previous consideration, the existence of two vanishing cycles with the
intersection number $\pm 2$ ensures the non-properness of the Lyashko–Looijenga
map, and hence the fact that the $\mu=const$ stratum of the singularity is
positive-dimensional.

Problem 1D.

Give more general lower bounds of the
dimension of $\mu=const$ strata in terms of intersection forms of vanishing
cycles.

That is, if we can indicate many independent prohibited collisions of critical
values, then probably the attempt to perform these collisions by brute
force will throw the corresponding Morsifications out of the neighborhood of the origin in ${\mathbb{C}}^{\mu}$ in independent directions (all of which approach the $\mu=const$ stratum).

The real versions of these problems are important for the construction of real
decompositions and enumeration of topologically distinct Morsifications of real
singularities: see [Chislenko1988], [Vassiliev2002b]. Namely, let $f:({\mathbb{C}}^{n},{\mathbb{R}}^{n},0)\to({\mathbb{C}},{\mathbb{R}},0)$ be a real function singularity, and $F\colon({\mathbb{C}}^{n}\times{\mathbb{C}}^{k},{\mathbb{R}}^{n}\times{\mathbb{%
R}}^{k},0)\to({\mathbb{C}},{\mathbb{R}},0)$ be
its real deformation. The space ${\mathbb{R}}^{k}$ of real parameters is
separated into several chambers by the real total discriminant (consisting of
all non-Morse functions and functions with critical value 0). We can go from
any chamber to any other one by a generic path in ${\mathbb{R}}^{k}$ passing only finitely
many times the discriminant at its non-singular points. Any such passage changes
the topological type of the function $f_{\lambda}$ in some predictable way.
Moreover, if our singularity $f$ is simple and the deformation $F$ is versal,
then all standard changes satisfying some natural restrictions can indeed be
performed. In particular, if $f_{\lambda}$ has two neighboring real critical
values, then we can collide them and get two critical points on the same level
(if the intersection number of corresponding vanishing cycles is 0) or
a critical point of type $A_{2}$ (if this number is $\pm 1$). In the
latter case, these two critical values (and the corresponding critical points)
go to the imaginary domain after this passage.

For non-simple singularities, we can perform all the same formal surgeries over
the collections of critical values (supplied with the intersection matrix and
some additional set of topological invariants of a real Morsification), and
combine these formal surgeries in arbitrary sequences.

Problem 1E.

What are the obstructions to the
realization of these chains of formal changes by paths in the parameter space
${\mathbb{R}}^{k}$?

An algorithm enumerating all such chains of surgeries was realized in
[Vassiliev2002b]. This algorithm was executed for all singularities of corank 2 and $\mu\leq 11$, and this execution never met a formal surgery which could not be realized by a surgery of functions in
the versal deformation.

Consider a one-parameter family of real analytic functions (or
just polynomials) $f_{\tau}:({\mathbb{C}}^{n},{\mathbb{R}}^{n})\to({\mathbb{C}},{\mathbb{R}})$,
$\tau\in(-\varepsilon,\varepsilon)$ realizing a Morse birth
surgery: the functions $f_{\tau}$, $\tau<0$, have two complex
conjugate critical points which collide in a point of type $A_{2}$
when $\tau$ approaches 0, and after that reappear as two real
Morse critical points of some two neighboring Morse indices.

Problem 1F.

Is there any convenient topological
characteristic of the function $f_{-\varepsilon}$ which allows to predict
these indices?

The parities of these indices can indeed be predicted. Namely, consider
the complex level manifold $V_{a}=f_{-\varepsilon}^{-1}(a)$, where $a$ is
a real non-critical value between complex conjugate critical values of
$f_{-\varepsilon}$ which are going to collide, and vanishing cycles in this
manifold defined by segments connecting $a$ with these critical values. The
intersection number of these cycles is equal to $\pm 1$ depending on the choice
of their orientations. Let us choose these orientations in such a way that the
complex conjugation in $V_{a}$ takes one of them into the other one. Then the sign of
their intersection number is well-defined and allows us to guess the parities of
the indices of newborn critical points: see [Vassiliev2002a]; [Vassiliev2002b]. But how can
we predict the integer indices?

Given a surjective map of topological spaces, $p:X\to Y$, the covering
number of $p$ is the minimal number of open sets covering $Y$ in such a way
that there is a continuous cross-section of $p$ over any of these sets. This definition
was given by [Smale1987] in connection with the problems of
complexity theory. In the particular case of fiber bundles, this notion was
earlier introduced and deeply studied by [Schwarz1966] under the
name of the genus of a fiber bundle. However, in the complexity theory
of equations over real numbers, the case of maps with varying fibers becomes
essential. Here is one of the first examples. Consider the 6-dimensional real
space of pairs of polynomials $(f_{a},g_{b}):{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$, where $f_{a}(x,y)=x^{2}-y^{2}+a(x,y)$, $g_{b}=xy+b(x,y)$, and $a(x,y)$ and $b(x,y)$ are arbitrary
polynomials of degree $\leq 1$. Obviously, the system $\{f_{a}=0,g_{b}=0\}$ always
has 2 or 4 solutions in ${\mathbb{R}}^{2}$ (counted with multiplicity).

Problem 2A.

What is the minimal number of open
sets $U_{i}$ covering ${\mathbb{R}}^{6}$ such that for any $U_{i}$ there is a continuous map
$\varphi_{i}:U_{i}\to{\mathbb{R}}^{2}$ sending any pair $(a,b)\in U_{i}$ into some solution
of the system $\{f_{a}=0,g_{b}=0\}$?

In previous terms, this is a question about the covering number of the
projection map $X\to Y$, where $Y={\mathbb{R}}^{6}$ is the space of parameters $(a,b)$,
and $X\subset{\mathbb{R}}^{6}\times{\mathbb{R}}^{2}$ is the space of pairs $((a,b),(x,y))$ such
that $(x,y)\in{\mathbb{R}}^{2}$ is a root of the system $\{f_{a}=0,g_{b}=0\}$.

The number in question is not less than 2 (indeed, we can emulate the complex
equation $z^{2}=A$ inside our system, and the covering number of this equation
depending on the complex parameter $A$ is equal to 2). But is this estimate
sharp?

Problem 2B.

The same questions concerning the
approximate solutions. That is, for any $i$ and any $(a,b)\in U_{i}$, the value
$\varphi_{i}(a,b)$ should be not necessarily a root of the system $\{f_{a}=0,g_{b}=0\}$, but just a
point in the $\varepsilon$–neighborhood of such a root for some fixed positive
$\varepsilon$.

These problems have obvious generalizations to polynomial systems of higher
degrees and different numbers of variables. They can be non-trivial even for polynomials in one real variable: see [Vassiliev2014].

The essential ramification set in the space ${\mathbb{R}}^{d}$ is the
union of all values $a=(a_{1},\dots,a_{d})$ such that the corresponding
polynomial $f_{a}$ has either a real triple root, or a pair of complex conjugate
imaginary double roots: see [Vassiliev2011]. Obviously, this set is a subvariety
of codimension 2 in ${\mathbb{R}}^{d}$.

Problem 3.

Is the complement of the essential ramification set in ${\mathbb{R}}^{d}$ a
$K(\pi,1)$-space?

This is actually the “odd-dimensional part” of Arnold’s problem 1987-14
from [Arnold2004] (repeated as problem 1990-27). I describe below its
reduction to a problem in algebraic geometry.

Any compact domain in ${\mathbb{R}}^{n}$ defines a two-valued function on the space of
affine hyperplanes: the volumes of two parts into which the hyperplane cuts the
domain. If $n$ is odd and the domain is bounded by an ellipsoid, then this
function is algebraic (by a generalization of Archimedes’ theorem on sphere
sections).

Arnold’s problem (see [Arnold2004]). Do there exist smooth hypersurfaces in ${\mathbb{R}}^{n}$ $($other than the quadrics in odd-dimensional spaces$)$, for which the volume of the segment cut by any hyperplane from the
body bounded by them is an algebraic function of the hyperplane?

Many obstructions to the algebraicity of the volume function follow from the
Picard–Lefschetz theory, studying the ramification of integral functions:
see [Vassiliev2002a], [Arnold and Vassiliev1989]. These obstructions are quite different in the
cases of even or odd $n$, because the homology intersection forms, which are a
major part of the Picard–Lefschetz formulas, behave very differently depending
on the parity of $n$. In particular, the “even-dimensional” obstructions are
sufficient to prove that the volume function of a compact domain with
$C^{\infty}$-smooth boundary in ${\mathbb{R}}^{2k}$ is never algebraic: see [Vassiliev2015].
Here are two similar obstructions specific for the case of odd $n$.

Definition.

A non-singular point of a complex algebraic hypersurface is called
parabolic, if the second fundamental form of the hypersurface
(or, equivalently, the Hessian matrix of its equation) is degenerate
at this point. A parabolic point $x$ is degenerate, if the
tangent hyperplane to our hypersurface at $x$ is tangent to it at
entire variety of positive dimension containing the point $x$.

Proposition.

(see [Vassiliev2002a]). If $n$ is odd and the
volume function defined by a bounded domain with smooth boundary in ${\mathbb{R}}^{n}$ is
algebraic, then the complexification of this boundary cannot have
non-degenerate parabolic points in ${\mathbb{C}}^{n}$.

Smooth algebraic projective hypersurfaces of degree $\geq 3$ always have
parabolic points (and moreover, by a theorem of F. Zak, they have only
non-degenerate parabolic points). Unfortunately, this is not sufficient to give
a negative answer to the above Arnold problem, because

(a)

the complexification of a smooth real hypersurface can have singular points
in the complex domain, and non-smooth hypersurfaces of arbitrarily high degrees
can have no parabolic points: for instance, this is the case for hypersurfaces
projective dual to smooth ones;

(b)

the previous proposition does not prohibit parabolic points in the plane at infinity ${\mathbb{CP}}^{n}{\setminus}{\mathbb{C}}^{n}$.

However, the standard singular points, which can occur instead of parabolic
points, the generic cuspidal edges, also prevent the algebraicity of the
corresponding volume function: see [Vassiliev2002a], §III.6.

Problem 4.

Are these geometric obstructions sufficient to solve the above problem?

In other words, is it true that the complexification of the smooth algebraic
boundary of degree $\geq 3$ of a compact domain in ${\mathbb{R}}^{n}$ has always a point of
one of these two obstructing types? If not, probably we can complete this list
by some other singularity types, which also obstruct the algebraicity, so that singular points of at least one of these types will be unavoidable on
any such hypersurface.

Given natural numbers $d$ and $N$, consider the space $P(d;N)$ of all smooth
algebraic hypersurfaces of degree $d$ in ${\mathbb{R}}^{N}$. The trivial elements of
this space are the empty manifolds if $d$ is even, and the surfaces isotopic to ${\mathbb{R}}^{N-1}$ if $d$ is odd. Consider also some natural measure of
topological complexity of such hypersurfaces, which takes the smallest value on trivial objects only: for example, the sum of numbers of generators of
homology groups, or the lowest number of critical points of Morse functions.

Problem 5A.

Is it true that any hypersurface from the space $P(d;N)$ can be
connected with a trivial one by a generic path in this space in
such a way that it experiences only Morse surgeries, which
decrease this complexity measure?

In other words, does our space contain non-trivial varieties with
the following property: any surgery of this variety decreases (or
leaves unchanged) this complexity measure?

This problem can be extended to algebraic submanifolds defined by systems of
polynomials; however, the measure of topological complexity in this case should take into account the possible “knottedness” in ${\mathbb{R}}^{N}$.

Problem 5B.

A version of the previous problem,
in which the complexity measure is not purely topological: namely, it is the lowest
number of critical points of Morse functions defined by restrictions of linear functions ${\mathbb{R}}^{N}\to{\mathbb{R}}$ to our varieties. $($Correspondingly, the
surgeries of the variety affecting this measure should be not only those of topological
nature but also include bifurcations of the dual variety$)$.

If the answer to the previous questions is negative, then we obtain functions
that associate with any value $T$ of topological complexity the lowest number $F$
such that any surface of complexity $T$ can be connected with a trivial one by
a generic path in the space $P(d;N)$ for which the complexities of all
intermediate hypersurfaces do not exceed $F$.

Problem 5C.

Give an upper bound for the function $T\mapsto F$.

It is appropriate to mention here a related problem due to V.A. Rokhlin.

Problem 5D.

Do non-singular real plane projective curves of an odd degree consisting of a single connected component form a connected set (i.e., are they rigid isotopic)?

Let $f:({\mathbb{R}}^{n},0)\to({\mathbb{R}},0)$ be a function germ with $df(0)=0$ and the Milnor number $\mu(f)$ finite.
Let $\rho(f)$ be the smallest number of real critical points of real
Morsifications of $f$.

Problem 6A.

Is it true that any real
Morsification of $f$ can be connected with one of complexity $\rho(f)$ by a
generic path in the base of a versal deformation, in such a way that all Morse
surgeries $[A_{2}]$ in this path only decrease the number of real critical
points?

Problem 6B.

What can be said about the number $\rho(f)$?

Two obvious lower estimates of this number are (a) the index of $\mbox{grad}f$ at 0,
and (b) the Smale number of the relative homology group

$H_{*}(f^{-1}((-\infty,\varepsilon]),f^{-1}((-\infty,-\varepsilon]))$

(1)

(i.e. the rank of the free part
of this group plus twice the minimal number of generators of its torsion). Of
course, the first estimate does not exceed the second one, but can they be
different? Do they coincide at least for functions of corank 2?

Can the group $($1$)$ have non-trivial torsion? Is the estimate
(b) of the number $\rho(f)$ sharp?

Problem 6C.

Is it true that any component of the complement of the discriminant variety of a versal deformation contains a Morsification, whose all $\mu(f)$ critical points are real?

Consider a polynomial ${\mathbb{C}}^{1}\to{\mathbb{C}}^{1}$ of degree $n$ and some simple root
$z_{0}$. Let $d$ be the minimal distance from this root to all other roots of
this polynomial. According to [Reshetnyak1962], the $\frac{d}{2n-1}$-neighborhood of
$z_{0}$ belongs to its domain of convergence; that is, Newton’s method starting
from any point of this neighborhood converges to $z_{0}$. The estimate $\frac{d}{2n-1}$ cannot be
improved as a function of $d$ and $n$ not depending of the polynomial.

Problem 7.

Give a similar universal estimate of the
radius of convergence for multidimensional Newton’s method of
[Shub and Smale1993].